A "change of basis" is an action performed in linear algebra, whereby a change in fundamental structure yields an entirely new viewpoint. This blog began as a record of a pedagogical change of basis for me, and continues as an ongoing account of my thoughts as I design and direct courses in mathematics at the University of North Carolina, Asheville.

Thursday, September 28, 2006

It's a take-home exam with four questions on it (three of which are divided further into subquestions). There are some computations involved, but nothing too heinous, and nothing that can't be facilitated by the use of a calculator or Mathematica.

There are a number of applications considered, and though a couple of them are novel, they're clearly laid out, and quite simple. The other applications are similar to others considered in class. I think the exam will test interpretation of computations better than it will test anything else. So you know what the rank of a matrix is? Great! Now how do you use it?

I've also written up the plans for tomorrow's class, where we'll consider one more application, this one related to my research, before we play a game of "Choose Your Own Matrix" in order to flex our rank-finding muscles.

On deck for Monday: vector spaces! Fun! And on Wednesday, we might have some guest teachers...

I gotta be honest: I'm beginning to think I'm manic-depressive when it comes to this course.

At one moment, I'm on top of the world, and at the next, well...maybe I'm just blowing the little things out of proportion.

What do you think?

At the end of the problem session this last Monday night, I was practically euphoric. We had a good turnout, we had fun, as usual, we (and I'm including myself in that "we") learned a lot, we made a good deal of progress.

And in class today, when one of the applications "clicked" with someone who'd been struggling with it for a long while, well...those "clicking" moments are the moments that teachers live for.

And of the five people I've now approached about leading discussions and activities in the class, all five are excited about it and willing to rise to the challenge. You have no idea how much this encourages me.

But I know that some folks are concerned that we're not moving fast enough...

...and some folks are concerned that we're moving too fast.

Some folks are concerned that they're not getting the "formulaic" aspect of the course, laid out in theorems and proofs and computations galore by the textbook...

...and some folks are concerned that they just can't see the pictures that we use to describe the applications that come up in class.

HOW DO I MAKE EVERYONE HAPPY?!!!?!!

I know the answer.

The answer is: I can't.

And that...that weakness...that inherent and unavoidable failure...gets to me.

I just can't make sure that everyone's completely satisfied. No one can do it. It just can't be done.

I guess the best I might be able to manage is to "maximize" satisfaction.

But how on Earth can I do that?

I realized something the other day about the way I've designed this course. On the one hand, it's a good thing that I've taken the trouble to create such robust methods for communication with the class: I've had no trouble developing a rapport with the students which I think is often underrated, I've been able to quickly and (I hope) accurately assess when people are having difficulties, and I've therefore been able to remediate problems before they get too nasty. I think that I've been able to do this far more effectively than I would have had I been teaching the course in a "traditional" way. And I'm not saying that's a bad thing. But...

...I also feel like ignorance could be bliss. Bliss for me, at any rate. If I cut all these lines of communication, I would no longer know right away if something were amiss...for instance, the way things are set up now, I can tell almost immediately if someone's having difficulty with a particular concept or example, and I can take steps to fix that as soon as possible.

The "old" way, someone could fail an exam before I'd even know what was up. And in all that time, during which that poor student was struggling, maybe faking his or her way through the homework and in-class examples, managing a mediocre but not pathetic grade on the quizzes, I'd be totally unaware of that struggle, and I'd assume all was well.

Maybe that's why I might be going more slowly through this material than I should be: I've got my fingers on the pulses of every body in that class, and if even one of those pulses starts racing, I hit the emergency brakes, and we slow down.

I know I shouldn't be doing that, I know I can't expect everyone to do equally well, but how can I do otherwise? How can I let myself fail in my role as a teacher, if I know someone's not understanding?

HOW?

I don't know.

So here's my plan for the next couple of days: we'll get that quiz in on Friday (at last!), and then we'll do one more example before we call it quits for Section 2.1. Section 2.2 will take us on a short tour of notation and terminology, and then, after I've handed out the take-home exam, we'll start some really cool stuff on Monday, when we take on arbitrary vector spaces. The applications are very quickly going to become much more interesting, more realistic, more varied in nature.

What do you think? How are we doing, as a class? How are you learning, as students? And how am I helping in facilitating that learning, as the, for lack of a better term, "teacher"? I really want to know.

Tuesday, September 26, 2006

So just guess which branch of mathematics played a role in the pure mathematical research I was working on this afternoon?

Hmm? Any guesses? I'll give you three, and the first two don't count.

Here's the problem. I've got a set of distinct integers ai, 2 or larger, arbitrarily many of 'em, maybe something like: {2,4,5,8,13}. Now you get to pick any positive number a (doesn't have to be an integer) lying strictly between the smallest element of the set and the biggest. In our example above, I could take 5.4, maybe.

The question is: how many different ways can I pick "weights" x1, x2, x3, and so forth, one for each element of my original set, so that each weight lies on [0,1], the sum of the weights is 1, and the sum a1x1 + a2x2 + ... +anxn is the number a? Above, for instance, we'd get to pick 5 weights, and we'd have to get the sum from the previous line to add up to our chosen value of 5.4.

I'll leave it as an exercise for the reader (a common math ploy!) to provide the details, but I'll cut to the chase: you can turn this problem into a linear system, and from there into a matrix, with 2 rows and as many columns as you have numbers ai in your set. Since you can have arbitrarily many elements in that set, you can end up with boatloads of columns (yes, "boatloads" is a technical term). And as soon as you've got more than 2 columns...what happens? I'll let you work it out.

Cool. This has some pretty heavy-duty implications in the land of graph theory.

I was so excited about figuring this out this afternoon that I went across the hall to the Math Lab, found Kaytlynne and Theophila working hard on homework from their other courses, and bugged them about the cool way linear algebra had worked itself into my work.

Neat, huh?

Tomorrow we'll get a few more folks from the class up at the board. Nadia and Imogene did a great job in getting us started on mapping out crystal structures. We'll pick it up from there.

Monday, September 25, 2006

I felt that class came off pretty well today. We began with an example of linear algebra applied to chess, in which we analyzed the moves of the knight in terms of the basis {[1,2],[2,1]}. Fun galore! From there we spent some more time on the applications begun last week, to crystal structure and roving bases that track the structure of a tropical storm.

And at tonight's problem session, we got our attendance back up to 11, with some new faces (and the sorely-felt absence of a few familiar folks), and with the SAME six teams represented as in the past two weeks. (Come on, you folks in the other two teams...what's keepin' ya?) Having now received permission from those depicted to post the pictures from tonight and last week, I've included (below) candid shots of the evening activities.

Also, I've approached four of the students from the class with the idea of letting them have the chance to lead the classroom activities, and all four were up for it! (I'm doing a little dance of jubilation as I write this. Don't believe me? Ha! Try and prove it!) All seemed excited by the idea, though one of the stalwart students expressed a little fright, which would be mitigated by being allowed to lead class with the assistance of a good friend. I have confidence in all four of these students' abilities, and will soon be prepping them for this responsibility. I'm excited!

Meanwhile, here are the promised pix from tonight and last Tuesday:

Okay, y'all who haven't come yet: see how much food you're missing out on? A veritable smorgasbord of goodness! Two sorts of apple bread (store-bought and homemade), homemade brownies (thanks, Fiona!), grapes, and your choice of refreshing carbonated beverage! Mmmmmm...MMMM!

And check this out: can you get any more excitement than the Row Reduction Races?!!? I mean, really! Look at these tough competitors from Week 2, pounding out those row operations on their calculators, each one hoping to be crowned the monarch of matrix manipulation!

And here it is! The exultant victor! Just look at that reduced row-echelon form! Yowza! It's this sustained, unceasing excitement that led to tonight's increased crowd, shown below:

Food, friends, linear algebra...a natural combination. I ask you, does it get any better than this?

Friday, September 22, 2006

This week's been rough on a lot of folks. I know of several tough classes in which there were exams this past week. I've seen a number of glazed-over looks and catatonic stares, and attendance has been low in all of my classes, especially today.

Friday. Wonderful, wonderful Friday.

It's been a rough week on me, too, I have to admit. Two fifteen-hour campus days started things off, followed by three only-slightly-shorter 11 hour romps. Mathematical action from dawn until dusk. I'm glad I've got an insuperable supply of energy, I've had to draw on it this week.

I knew I'd be putting a lot of effort into this class. I'd tried to front-load it as much as possible by doing a good deal of the planning before the semester began, and to a large extent that planning's paid off: I've had less to do on the fly than I would have had I not put together beforehand a goodly portion of the projects and class activities, much of the documentation, and a boatload of resources. Still, though, there are some midcourse adjustments to be made, a few wrongs to be righted.

What's gone wrong so far?

Nothing big, really. Minor missteps, here and there.

For instance?

For instance, I've been overestimating how much we can "cover" in a single class period. Given the more open, bidirectional format of the course, the exchange of information is certainly not as "efficient" as it would be were I standing at the board lecturing the whole time. I'm just now getting a feel for how many in-class exercises we can cap off before Doctor Bob's unofficial class clock strikes 3:35ish.

I also underestimated the difficulty some folks would have in reading the textbook. I know it's a dense read, my friends. Most upper-division math texts are, especially the first time around. I'm thinking back to the first math text I had to really read (Richard Strichartz's The way of analysis, the basis for my first two semesters of real analysis): though I now find it a better-than-average text for its topic, quite funny in places and very lucidly written, I know at the time that I was first reading it it might as well have been written in Sanskrit.

Nevertheless, the text is a decently good one, as linear algebra texts go. It does a fair job of clearly developing the necessary theoretical points while saving time for more mundane computations. And I hope the "Key Points" slides I've been providing in class have softened the textbook's blow, and that our time in class has helped to clarify any obscurities encountered in the reading.

If it's any consolation to my students, reading math doesn't necessarily get any easier: you can't imagine how many times I have to read every page of some papers in order to understand them as well as I'd like to.

Still, as hard as the reading can be, it's crucially important that every one of us does her or his best to keep on top of it. Quite frankly, the in-class exercises (which are meant to be challenging, but not heinously so) will be much more enjoyable for those who've done the reading and have prepared themselves for class. Should one expect to do at all well in a literature seminar centered upon a certain novel if one has consistently gone to class without first reading the book? I hope the answer to that question is "no."

All in all, I think that I'm doing just about all that I can to make things go smoothly.

What more can I do? So much of the class is up to the students, I can only go so far.

Well.

Well, well.

Well, I'm off to read a few of the journals submitted on-line.

Tonight, and then tomorrow, and then...and then tomorrow again. Then we'll start anew.

Yes, I'm tired.

More to come.

Meanwhile, let me throw out a question to my students: how many of you out there in linear algebra land would like a shot at leading the class discussion for ten or fifteen minutes? What would you do with that time if I gave it to you to use? I really would like to know, feel free to write!

Tuesday, September 19, 2006

Tonight's showing was a little sparser, largely, I would guess, on account of the heavy exam schedule burdening a lot of folks this week. I know there was a crippling Organic exam yesterday, and there's a Complex exam tomorrow in our own department. Lots goin' on, and people are having to divide their time.

For the few who came tonight, there was much food and much fun. There are pictures to post, but as usual I want to get permission before posting them.

In all, eight folks showed up, representing 6 teams. Good spread! I feel like we got A LOT done tonight. Y'all are going to be experts by the time this semester is over. Good work! And keep it up...

Sunday, September 17, 2006

Conversations with a few of you folks over the last couple of days, both on this blog, and over e-mail, have made me think a lot about pulling off a good research project.

For those of you who haven't been reading all of the comments lately, someone responded to my post from yesterday, anonymously encouraging me to light a fire under y'all's behinds and get you going on those research projects. In reply to this, I wrote another comment giving some tips on how to get a good start on these projects, and how to sustain them well as the semester progresses.

And then, while writing back and forth with another member of our class about undergraduate research, I got to thinking: hey, since when am I the sole font of knowledge in this classroom? Many of you have already done (and perhaps are doing) undergraduate research, and I'm sure you have lots of good ideas about how to start off a good research project, how to go about keeping track of your sources, how to record your ideas and later write about them...and so on.

I'm not claiming that my hints won't help, I'm sure they will. But I'm sure that many of you have lots of other hints you could share with your peers in this class, to make the ride a bit smoother.

So, I'm kindly asking all of you who've had some experience in research before: what's helped you out? Do you have any secrets to impart? Ideas you'd like to share, particularly, at this time, regarding how to get started? I'd greatly appreciate if you'd chime in by responding to this post, even if you do so anonymously.

Saturday, September 16, 2006

It's Saturday, and I'm just now fixin' to put together a solution sheet for the latest round of representative problems, as I've taken to calling the recommended textbook exercises for 365.

I've spent the afternoon up until now working on a statement of my teaching philosophy, an exercise which meant several hours of agonizing over the right choice of words. Teaching is something I could easily prattle on and on and on and on and on about for chapter after chapter (surprise, surprise), so a good deal of the difficulty lay in cutting what I had to say down to the requested three pages. I was forced to think about the elements of teaching and learning which mean the most to me.

Hmmm...

I'll think some more about that matter. What's on your mathematical minds right now?

In the meantime, I wanted to put out a reminder for my students in 365: please don't forget that Tuesday night at 6:00 we'll have our second problem session, location TBA. I'll be sure to bring a loaf of my homemade apple bread (mmmmmm! apple bread...), and I've already heard murmurred rumors of brownies and soft drinks from some of the students. Please feel free to bring snacks, music, beach balls...whatever will make the session more helpful, more comfortable, more fun. No matter what else you bring, bring yourselves, and let's see if we can break last week's attendance mark of 11!

Wednesday, September 13, 2006

Today saw us going over matrix inverses again, more fully than we'd done on Friday. I felt that Friday we'd not yet had a chance to really get a good grip on the geometric meaning of an inverse, so we retraced our steps into that territory before learning how to run a Markov process in reverse.

Although I think most of the folks in the class got the hang of it by the end of the class, there were a few who remained confused. (One student came by after class, frustrated as hell at not picking up on a lot of what we'd done. We spent about fifteen minutes working through the applications we'd considered, one-on-one. By the end I think it was clear that the student understood more of what was going on than the student had thought at first.)

Not knowing what in the wide world of wackiness is going on is truly frustrating.

And there's no doubt about it: frustration sucks.

Frustration works its ways differently on different people. Some cry, some scream, some just lose all capacity for rational thought.

The most frustrating incident of my academic career came in my third year of graduate school, at the end of the Fall 2000 semester at Vanderbilt.

That'd been a tough semester. If memory serves, I had a somewhat traditional algebraic topology course, a couple of seminar classes, including one on combinatorial group theory and another on small cancellation theory, and I was auditing a class on advanced group theory. Meanwhile I was teaching a pretty cool Calc I class, helping to put together our department's undergraduate seminar in mathematics and compose our in-house precalc review text, and conducting research that would later become my first published article.

By early December, I was pretty much gone.

The seminar on small cancellation theory was an exciting one. The instructor, one Professor Peanut, was (and is) one of the world's foremost experts on combinatorial group theory, universally respected in his area, and well-known outside of mathematics for his theories on the use of mathematics in deciphering biblical prophecy. Big, big man in the mathematical community. A very kind gentleman, he terrified me nonetheless with his stature and renown.

He was a man of deep conviction and numerous idiosyncrasies. An orthodox Jew, he davened constantly while reading his Torah, something he did during just about every break available to him. He played with his beard absentmindedly, stroking its great gray length with thick stubby fingers. He drank Diet Coke as though he owned stock in it. If he were not the one speaking but only participating as an audience member, he would sit in the front row (an open can of Diet Coke in front of him), his attention fixed on the speaker for all of about ten minutes before sleep overtook him and he dozed off, his chin sinking through his beard to rest upon his chest. He would sleep for five or six minutes at a time before waking and looking around as though to make sure no one had caught him in his nap. Five minutes later the cycle began anew.

Strange travel restrictions forced him to spend with us only half of the semester he'd promised our department, so it was not until mid-October that he came to the United States to begin the seminar course we'd scheduled for the fall. Once underway, we (the five of us registered for the course and a few faculty hangers-on who wanted to sit in) decided the best way to make up for lost time would be to meet once a week for twice as long as we'd originally meant to meet. So began our three-hour Thursdays.

The class was fascinating. Peanut would lecture to us extemporaneously on cutting-edge research, much of which he'd developed only weeks or even days before. Unlike his long-time colleague and co-author, Professor Cashew, who for the past year had been a full-time faculty member at Vanderbilt, Peanut was not a slave to detail and would almost invariably omit precise computations and rigorous proofs, opting instead to give outlines and general descriptions of his intricate arguments. Though he painted in broad strokes, the pictures he produced seemed clear to us, and only occasionally would we have need to stop him in his exposition. Much of the beauty of math lies in the creativity that goes into the construction of new techniques, and Peanut's were among the most delicate and delicious I've ever seen. Though his exposition was too sparse to allow me to get at the innermost workings of his mathematical thought, I felt I had a good understanding of the subject.

I realize now how difficult it is to learn to paint without actually getting the chance to hold the brush yourself.

We spent nine or ten evenings in these seminars, and in that time I filled perhaps four dozen pages with notes on Peanut's techniques. Supplementing this were about thirty pages of poorly mimeographed notes Peanut himself had written (by hand, in very nice and readable Russian script) in preparation for writing one of his papers. We had a good written record of the course, but given its lack of precision (like his speech, my notes are filled with phrases like "quite big," "big enough," "very small," and so forth), to do much math with what we had would be like to building a warp drive from pencils and chewing gum after watching an episode of Deep Space Nine.

I must take a moment to describe one of the course's faculty fans. Professor Pistachio was (and is) a brilliant man with an acerbic wit and a certain impishness that came out most noticeably when torturing undergraduates. On his website he keeps a page which admits that his students often complain he is too sparing with positive feedback. To counter this, he includes on the page a MIDI file of soothing music, after several seconds of which he intones carefully and lovingly, "good job. Good job. You like the word 'great' better? Great job..."

"Sarcastic" doesn't cover it. (In his defense, I must say that I consider Pistachio one of my favorite professors in graduate school, an excellent expositor of mathematics, and one of the best writers of mathematics I've ever known: it's hard to write clear technical mathematics articles, and he does this exceedingly well.)

At the end of the last class, Peanut dismissed us. "That is all," he said, and with a kindly smile he set down the chalk. If only it were so! From his seat to my left, Professor Pistachio said, in Russian, "what about their exam?"

We sank. We'd expected to not be tested; it was an unwritten understanding. After all, this was a seminar course, and those who participated did so because they wanted to learn the material. We weren't undergrads anymore, we didn't have to be prodded and cajoled by some penny-ante system of praise and punishment. We didn't need an exam!!!

More to the point, we weren't ready for an exam. What would he, could he ask of us? How could we be tested on details we'd never learned?

"Ah yes," Peanut replied softly. "There is the matter of this exam. I think perhaps it should be...oral examinations. You will each schedule a time to meet me individually, yes?"

We left the room dejectedly. I trudged back to our basement offices with my colleagues Damon, Maurice and Nathaniel. We discussed our plans. "We've got a week," we agreed, "before he leaves the country. We can study. We have notes."

But we were all busy. We had finals to write and proctor and grade, we had research, we had other classes. Our time was precious, and we could spare little of it to digest the abstruse methods we'd been given in big, unchewable chunks over the preceding weeks.

"I'm going to do it tomorrow," I told my friends the next day.

"Daring," they agreed.

"I just want to get it over with," I said. I sent Peanut an e-mail and made an appointment at 4:00 the following afternoon.

I spent the next 24 hours cramming like never before. I read over my notes again, forwards, backwards, sideways, diagonally. Like Peanut with his biblical gematriyot, I sought meaning in every word.

4:00 the next day came way too quickly. I knocked timidly on the open door of the office Peanut shared with a few other visiting faculty members. There was no answer. I knocked again, and again there was no response. I padded into the office softly and peered around the cubicle divider which separated Peanut's desk from the one nearest to the door. At his desk he sat, squinting at the screen of his laptop. "Professor Peanut?" I asked. He looked up and smiled and gestured for me to sit in the chair across from him. I sat, and the test began.

"First, can you tell me how to describe the automorphisms of the...icosahedron?"

What in the HELL?!!?!!? This question had NOTHING to do with ANYTHING we'd talked about in the last several weeks. NOTHING. Nada. Zippo. My mind spun.

"You have had a basic course in group theory?" Peanut asked, his tone belying at least a hint of annoyance.

"Yes," I insisted. "But..." He scribbled a sketch and a hint or two on a piece of paper and thrust it across the desk towards me. I stared at it blankly. "Um...let me think..." And he did. He folded his fingers together and sat back in silence as I struggled to make connections between what I knew of icosahedra and what I'd learned in Peanut's course. There wasn't much to go on, but I managed to retrieve a few tidbits concerning fixed point sets and fundamental domains, and with these I pieced together a good bit of BS.

I looked up. Peanut, true to form, was asleep.

"Professor Peanut?" I asked gently. "Professor Peanut?" He slept on. I leaned back and sighed, and the hot light of the setting sun cut into the room through the slats of the office's Venetian blinds. I squinted and looked outside at the walls of the physics building next door. I wished myself far away.

I tried again to rouse Peanut, and at last succeeded. I showed him my notes, and he nodded and made a few ambiguous and uninterpretable noises. Was I right on or full of crap? I had no idea. As he let me get back to work, I looked wonderingly at the notes I'd produced. I'd been in Peanut's office for almost an hour and had done almost nothing. Quite frankly, I felt like shit.

Peanut let me struggle for another quarter-hour, and at the end he let me off the hook. "Okay," he said. "That is enough." I'd made a half-assed attempt at solving one easy question, and in producing that one solution I'd demonstrated minimal knowledge of a few minor techniques we'd covered in the class. The worst was yet to come.

"Now I must assign a grade," said Peanut. "I'm afraid I do not understand...what does this mean, an 'A'? What does a 'B' mean?" I gave him a briefing on the American grading system, perhaps curving each category a bit too generously. Peanut seemed satisfied. His next question came sock in the gut.

"What grade do you deserve?" he asked.

After the worst academic performance of my life, I'd been asked to assess its merit.

The lowest grade I got in grad school, I gave myself.

I left Peanut's office numbly and dumbly. I went to my office and let the others know how it had gone, and then I left. I spent the rest of the evening in a stupor, eating flavorless food and having mirthless conversations.

Yeah, frustration sucks.

I hope that if you're ever frustrated with this course, like our friend who spent some time with me this afternoon, you'll be willing to share that frustration with me. It feels better when it's spread around, and once it's out there, we can do something about it.

Okay, that's enough for tonight. I'm sure you've all enjoyed this bedtime story. Feel free to post away and share your own moments of frustration! Meanwhile, let's hit that reading for the weekend, and get ready for some hardcore Tinkertoy action on Monday!

We had the first of our after-hours problem sessions tonight, and I feel it went very well.

The session was a fantastic opportunity for all of us: while the students in attendance (11 of the class's 33, representing 6 of the 8 teams) seemed in complete agreement that the session helped to solidify their understanding of the underlying computations, I felt that I gained tremendous insight into the students' thoughts on the class overall. We had some refreshingly open and honest conversations about how people are feeling about the course right now, and I'm heartened. I love these people!

We'll have another such session next Tuesday night, but be forewarned: I fear I'm going to have to start asking for a couple of bucks a pop (or "a soda") to defray the cost of refreshments!

My mind is aflutter with thoughts, but sadly I'm too tired to track any one or two of them down to pin them to the page, so I'll leave it at that.

I will end with another exhortation to those who are reading this blog (and after tonight, I know there are quite a few of you!) to please feel free to post responses to my own posts.

Saturday, September 09, 2006

Maggie tells me it's too much on my mind. She's probably right, as she often is when it comes to questions about me. I hate to sound cliche, but truly she knows me better than I know myself.

I logged on hoping to have something to say, but I'm having trouble finding words to wrap around my thoughts.

Am I overthinking this course?

I'm enjoying myself immensely, but I still find myself feeling unsettled. I think much of this has to do with the wacky schedule UNCA's served up for us this semester: a long first week took the steam out of everyone's engines before the first break rolled around; Labor Day stumbled in awkwardly after two full weeks had passed, granting us a shortened week this past week; and next week we've got an artificially truncated schedule with the cancellations in honor of the installation of our new chancellor. We've not yet had a chance to find our groove.

With a handful of nonmathematical classes, 365's borne the brunt of this clumsy scheduling. We've already spent class time on the syllabus and technical writing, and this coming Monday will bring us to the library.

I certainly don't regret setting time aside for these topics, and when I next teach this course you can bet they'll still play a major role, yet it'll be nice to have a purely mathematical straightaway in front of us by the end of this coming week.

This coming Wednesday we'll take another look at matrix inverses. Given the importance the geometric interpretation plays in a number of the applications we'll consider later on in the course (and that these folks'll consider throughout their careers!), I want to make sure everyone's got them down. Giving another day to inverses has the added bonus of allowing those who are struggling with the reading to get caught up.

In each of his first two journal entries, one of my atmospheric science students threw down the gauntlet, demanding to know why I'd yet to bring out any atmos applications, given that 3/11 of the class are ATMS majors. In answer to his challenge, I insisted that most of the anywhere-close-to-realistic applications of linear algebra to the atmospheric sciences are too complex to use as in-class examples. I felt this answer was pretty lame, though, so I hit the library yesterday afternoon between classes and checked out three texts on numerical weather prediction and linear climatic models, and spent a few hours early this afternoon slogging through a chapter or two of one of them, learning about matrix methods in airmass modeling. Although the physical equations governing the models are indeed complicated ones, if they're treated as black boxes, there are reasonably simple applications involving as few as four equations which might be workable in class. I'll have to have a crack at it. (To my challenger, if you're reading this: howzabout them apples?)

Okay, I've had more to say than I'd thought I would. Maggie'll be calling soon, to get a ride home from work. I'd best be going.

She's probably right: I'm overthinking this course. But I'd rather overthink than underthink.

Friday, September 08, 2006

Last night I finished reading the first of my students' journal entries. Wonderful insights! These folks have some fantastic ideas, a couple of which I'm already beginning to integrate into the structure of the class.

For instance, beginning this coming Monday I'll be holding informal (but informative!) problem sessions on Monday evenings, for those who'd like the chance to work on the HW with each other in a setting conducive to study. We'll start out in the Math Study Room on the second floor of Rhoades, but if that place doesn't "feel right" for some after-hours linear festivity, we may pop over to some other campus location in the following weeks.

By and large people seem to be settling in and having fun. The two main concerns on most people's minds are (a) the readings, and (b) those darned articles! I think I might have introduced some sample bias on that second point, since the second journal entry solicitation specifically asked the students to write on the issue of the project: "How do you feel about...?" Regarding the first point, while a number of folks are taking the reading in stride and handling it nicely, some others are having trouble digesting the pulp and getting to the real meat of the matter. I sent a pretty detailed e-mail yesterday afternoon giving some more hints as to how to read the text more effectively. I hope that e-mail helps folks direct their studies and make better use of their precious time.

For the most part, though, people are enjoying the class, liking their teams, and getting comfortable with the format. Almost without exception people are having fun with the team activities and the classroom atmosphere, and people seem confident that their teams can handle the semester's rigors.

Today's agenda's got matrix inverses written all over it. After looking at the geometric interpretation of inverses, we'll find out how to make the Markov Dance run backwards.

Wednesday, September 06, 2006

Class went pretty well today, I thought. (As always, I'd be interested in hearing the student take on this assessment...) The team quizzes were great, the individual quizzes were better, and there was some serious row-reduction of matrices goin' on in our hands-on approach to chemistry. Due to the size of some of the matrices which arose from those chemical reactions (6-by-7 in some cases!), I don't think many people managed to get their reactions to balance through linear algebraic methods. (One notable exception was Deidre: while at first her coefficients didn't come out right, she soon caught her small error in arithmetic a few row operations back, and hey presto! the balanced reaction popped out. Props, Deidre, if you're reading this!)

Anyhow, things went well, and I'm happy to say that it looks like folks are really getting the hang of using matrices to solve linear systems.

A couple of folks have mentioned that it's hard to focus one's attention on the highlights of a day's reading without knowing exactly what those highlights are. To try to address this weakness in the course design, I've begun (with today's class) to provide a short "focus" session at the beginning of class, in which we spend about five minutes hitting the major points of interest from the reading for that day's class. Jeremiah recommended this particular idea to me this morning, and I think it worked out really well today. (Props, Jeremiah!) I'll try to make that a regular part of the class, and students should feel free to comment on how it went today, and how we can make it work better in the future.

On deck for Friday: matrix inverses. I plan on bringing in some geometric applications that might be of especial use to any of the folks whose projects involve geometry (Crystal Gazers and Screen Flickerers take note).

Sunday, September 03, 2006

(My partner in conversation below is a fictionalized amalgam of real people and of inner discourse. He is a cipher, into whom I've placed words as freely as he's spit them out.)

Ethelred: Why?

Me: Why not?

Ethelred: Why not? Because you can't possibly cover as much. I mean look: how many of your students are going to be able to state the Cauchy-Schwarz Inequality, let alone prove it? Now it's come and gone, and you've never said a word about it in class.

Me: How many of my students in a traditional Linear class would be able to state the Cauchy-Schwarz Inequality a week after it's "covered"? And how many of them will ever use it?

Ethelred: Okay, bad example. But what about the Triangle Inequality? How are they supposed to understand more general inner product spaces later on if you're not even making sure they get the tools they need to understand the most important one?

Me: They've understood that most important one all their lives, just have them try to draw a triangle where one side's longer than the other two put together. So what if they don't know "why" that must be?

Ethelred: Now you're being pedantic.

Me: Probably.

Ethelred: You have to admit that Friday's class didn't get as far as you'd hoped it would.

Me: Maybe. Maybe at first. Maybe at first I thought that we'd fallen a little bit behind.

Ethelred: Ah. Ha.

Me: At first.

Ethelred: And then?

Me: And then I thought, "you know, if we'd finished up that chemistry exercise, we'd have worked our way through the solution of a linear system by row-reduction of matrices. We'd have finished up Section 1.4. We'd have been ahead..."

Ethelred: So you're still thinking in terms of "coverage"?

Me: "Coverage" is a four-letter word.

Ethelred: But you don't deny that it's on your mind?

Me: I don't deny it. A decade of teaching more or less one way, even if that one way's got some bells and whistles on it every now and then, doesn't disappear overnight. But you interrupted me.

Ethelred: Sorry.

Me: "...and they would have pushed their way ahead on their own. By themselves. They would have worked out the application for themselves. I wouldn't have done a single example for them."

Ethelred: So what?

Me: So how would you have had me teach Sections 1.1 and 1.2? "Definition, theorem, proof, theorem, proof, incomprehensibly inapplicable theorem, unnecessarily dense proof, useless abstract example, definition, long and boring (however impressive-looking) list of properties...half of which everyone'll forget a week after the exam, all of which half of them will forget a week after the exam..."

Ethelred: You're being pedantic again. There's a lot to say for the traditional Linear Algebra classroom. Are you really doing your math majors a favor? They, at least, deserve to see worked-out proofs of the major theorems from linear algebra. They, at least, deserve to gain access to the more technical aspects of the material.

Me: They'll get that in the reading. Besides, what would you have me do, sacrifice the rest of the class for the sake of the math majors?

Ethelred: There are a good many of them in there.

Me: There are just as many, if not more, atmospheric scientists, and you can bet most of them could give a monkey's red rear end for whether or not the Cauchy-Schwarz Inequality holds in any given inner product space.

Ethelred: Whether they care about it or not, exposure to that fact is good for them. Damn it, Patrick, you've said it yourself all of these years: above all else, the best thing that a student can get out of a math class is a little exercise in critical thought.

Me: And who's to say I've changed my mind? Besides, the person who said that was a younger version of my self-as-teacher, an alpha model. An inchoate form, Patrick-of-five-years-ago. I've rested safely and soundly in my old teaching method for years now, and though it's worked for me well all those years, there was something missing.

Ethelred: What was missing? Patrick, you've always been a good teacher. Why change?

Me: I've always been good, but why not be better? And I'm not sure I've been as good a teacher as I should have been. "What's missing?" you ask. Fair question.

Ethelred: And?

Me: Substance. Structure. A coherent overall frame. A big picture.

Ethelred: Overrated.

Me: Really? You say I want to teach critical thought. Fair enough. What does that mean?

Ethelred: And?

Me: No, really, what does that mean, to teach critical thought? It's something I've taken for granted, something I've assumed myself capable of, something I've assumed I could recognize, that others could recognize, when they'd achieved it. It's something I've always thought inhered in mathematics, so fundamentally so that you couldn't get out of a math class without having exercised it at least somewhat. But that's not so: you can make it through Calc II without even so much as rubbing your elbows up against critical thought, though you both sat side-by-side in a crowded classroom the whole semester. The problem is that the students don't recognize critical thought. And I've grown so accustomed to thinking that Irecognize it that half of the time I don't, either.

Ethelred: What, so you have to tell them that they've had a "spiritual experience" in a math class in order for them to actually have one?

Me: Not necessarily, but you may at least have to set them up for one. I'm not sure I was doing that before, however much group work I was doing, however many cupcakes I got them to cut up, however much fun and interactive my classes might have been.

Ethelred: And what's so grand about this Linear class?

Me: The other day Livonia was in my office, hangin' fire, chewing the fat. We were talking about the class, and all of a sudden she said something like, "I have a feeling this class is really going to help me with my upper level math classes. It's hard to do, but I'm going to need to know how to learn math on my own."

Ethelred: So?

Me: I pulled up the syllabus on-line and scrolled down to the learning goals: "be aware of your own learning styles, and of how to make effective use of those styles."

Ethelred: Hmm.

Me: Just a few minutes later, she was talking about her team's research proposal, and she mentioned how important understanding of the issues involved in their top choice would be later in her career: "that's the kind of thing I'm going to have to do, take a problem and take it apart to understand it, and then put it together again." I pointed back to the syllabus: "dissect a real-world problem in order either to analyze the way in which linear algebra can be applied in order to solve the problem, or to explain why linear algebra might not be applicable."

Ethelred: She's a bright kid.

Me: Oh, yeah. You can bet dollars to donuts she is. That's a roomful of smart people I've got in there. Don't think I'm not afraid of letting them down, of not helping them to find the best damned education they can get.

Ethelred: And your list of learning goals the way to do that?

Me: It's not just a list. It's not just a touchy-feely component of a lifeless document that becomes irrelevant as soon as it's printed. It's the shadow of something far more important, of an underlying structure, of an edifice that I built before the class even began. That list is a continual reminder to myself of everything I've got to do in the classroom before I head over there at 2:35 on Mondays, Wednesdays, and Fridays.

Ethelred: And what do your students have to do? Do you think that they're thinking about these things, too? Haven't they got enough to think about, with all the reading they've gotta do, all of the writing, all of the crap you've got them doing in the class itself?

Me: I don't know. I hope I've set it up in such a way that they don't have to think about the list, that every one of those fifteen goals is met without them knowing it. That's my intention, anyway, and whether or not I'm succeeding...well...we'll see how it turns out. Edward was asking me the other day about teaching one of the freshman colloquia: did I think I'd want to do that at some time soon, did I have any plans, any ideas for such courses? I said that yes, I had an idea that I'd been kicking around for a while that would make a perfect LSIC course, but that I wasn't ready to teach it yet. "It's probably a pretty good idea to get settled into the department before branching out like that," he said. Or something to that effect. That got me thinking: is that what I'm doing? I don't think that it is.

Ethelred: You're not getting settled?

Me: No, that's not it. I think I've done a fantastic job of getting settled into the department. I don't think that I'll ever be done doing that, of course. In fact, you're probably in trouble if you ever start to think that you're done getting settled. I mean that I'm not just sitting around in the math department, waiting for someone to give me permission to take a step outside Rhoades-Robinson: 365 isn't just another math class, and I have to say I took a little offense at the implication that that's all it is.

Ethelred: Is that what he meant to say?

Me: Maybe, maybe not. But that's the way I heard it. This course is so much more than that. You want a writing intensive course? You got it. You want an information intensive course? You got it. You want a course that provides a legitimate research experience? You got it. You want interdisciplinarianism? You got it. It's a more integrative course than just about any LSIC course I could ever dream up. This course is the perfect example of everything UNCA claims to be about.

Saturday, September 02, 2006

Team One will be working away at "Makin' Waves," analyzing the structure of wavelets from a linear algebraic point of view. In particular, they might ask the question: what is MP3, and how does it work?

Teams Two and Five (the first of which, cobbled together from Atmospheric Science majors, has dubbed themselves Team "Mesoscale Matrix") have been assigned to study "Traffic Patterns," and will undertake an analysis of various ideas related to network flow. One of these teams even offered up the possibility of working to unsnarl traffic on Asheville's own Merrimon Avenue!

Teams Three and Seven will each tackle the topic of "Crystal Gazing," taking a look at the application of linear algebra to the structure of crystals. One team comes at the problem from the point of view of chemistry (Team Seven's made up of Chem majors), while the other adopts a decidedly more "artistic" viewpoint, coming from a humanities/pure mathematics background. I'm sure the end results of these teams' endeavors will differ considerably, despite having the same starting point.

Team Four, a heterogeneous group made up of future and current teachers, pure mathematicians, and atmospheric scientists, will develop a mathematical means of understanding the game of Monopoly. The three now-and-future educators in this team realize already the enormous pedagogical implications of understanding the math behind this game and others like it!

Team Six, another motley group of biochemists, math majors, and one talented person who's not yet decided which of these fascinating subjects she wants to pursue, has been assigned to watch some "Flickers on the Screen" in order to understand how linear algebra plays a part in manufacturing computer graphics.

And Team Eight, the second of two teams of Atmos majors, will grapple with "Waste Water and Whatnot," an investigation into the role of linear algebra in solving systems of differential equations, beginning with those that arise in problems involving mixing rates.

The proposals for these projects were uniformly good. Some very strong cases were made for team qualifications to pursue proposed projects, and each team did a splendid job in making clear its particular interests. The decisions on my part were for the most part easy ones, largely because of the hard work the teams put into their proposals.

A good job done by all: keep it up!

Meanwhile, some conversations I've had in the past 48 hours with students and colleagues alike have made me think more deeply than ever before about this course and its design, and have convinced me strongly that running the course the way it's being run was the right choice to make. I'll say more on this topic once I've got my thoughts in better order.

Finally, I'm pleased to see that students are in fact reading this blog, and are feeling free to post comments to my posts. I welcome this heartily, and I encourage you all (students, colleagues, friends) to keep reading and posting; I ask only, as I always do, that you remain civil and respectful of others. Thank you all for your understanding and cooperation!

Friday, September 01, 2006

...The 2006 UNCA Department of Mathematics Beard-Growing Contest began last night; a picture of the contestants (your humble narrator at the far left) is shown below. This is the first time I've been without facial hair for about five years, so it feels veeeeery strange...